1) On Presidents' Day, AJ took a drone to fly at a local state beach. He arrived early enough with his parents to avoid the crowds. i) Before launching the drone, AJ decides to make a small mountain of sand as a target on which to land the drone. He fills a bucket with wet sand, carries it to a spot away from the waves, and pours the sand in an approximately even stream. The sand forms a conical pile. He notices the cone's radius is always about twice the height. He notices the rate at which the height increases changes. From the volume of the bucket and the time AJ takes to empty the bucket, AJ estimates the sand is being dropped in the pile at rate of 10 in/ sec. How fast is the height increasing when the height is roughly 30 inches? (Recall that the volume V of a cone of radius r and height h is V = rh.) ii) After launching the drone in a direction perpendicular to the ground, AJ runs away from the launch site to get a better view of the drone's increasing height. When the drone is 120 feet above the ground, it is rising at a rate of 15 feet per second. At that same instant, AJ is 50 feet from the launch site and running at a rate of 5 feet per second. How fast is the distance between AJ and the drone changing? iii) After the drone reaches a height of 300 feet, AJ begins flying the drone in a figure eight pattern which can be described by (x + 4y) = 256xy, where the two loops cross directly above the launch site. When the drone is 2 feet east and one foot north from this crossing, a California brown pelican flies by at the same height and 5 feet directly east from this crossing. The drone has a video camera and it is turned on. The line of sight of the camera at any time is tangent to the path of the drone. Does the drone capture footage of the pelican at this instance? 2) The following weekend, AJ took a drone to fly at another local state beach. He again arrived early enough with his parents to avoid the crowds. i) Before launching the drone, AJ decides to make a small mountain of sand as a target on which to land the drone. He fills a bucket with wet sand, carries it to a spot away from the waves, and pours the sand in an approximately even stream. The sand forms a conical pile. He notices the cone's radius is always about three times the height. He notices the rate at which the height increases changes. From the volume of the bucket and the time AJ takes to empty the bucket, AJ estimates the sand is being dropped in the pile at rate of 12 in/ sec. How fast is the height increasing when the height is roughly 25 inches? (Recall that the volume V of a cone of radius r and height h is V = rh.) ii) After launching the drone in a direction perpendicular to the ground, AJ runs away from the launch site to get a better view of the drone's increasing height. When the drone is 100 feet above the ground, it is rising at a rate of 18 feet per second. At that same instant, AJ is 75 feet from the launch site and running at a rate of 4 feet per second. How fast is the distance between AJ and the drone changing? iii) After the drone reaches a height of 300 feet, AJ begins flying the drone in a pattern which can be described by (x + y) 4x12xy, where the three loops cross directly above the launch site. When the drone is 2 feet west and two feet north from this crossing, a California brown pelican flies by at the same height and one foot directly west from this crossing. The drone has a video camera and it is turned on. The line of sight of the camera at any time is tangent to the path of the drone. Does the drone this time capture footage of the pelican at this instance? =